should the notes be allowed to ring?

hey i just started playing the guitar and i was wondering, am i susposed to mute the string(s) after i play a note or chord?

for example if i play 3rd fret 1st string then play 3rd fret 3rd string, do i play the 1st string 3rd fret, then release the 1st string to stop it from sounding, then play the 3rd string 3rd fret? or just hold my hand position and let both strings ring togetheras i play?

If i am susposed to release/mute after playing each note/chord how would i mute an open string such as playing open 1st string then 3rd fret 3rd string? i wont hav any fingers on the open string so how do i 'release' the string to mute it?

It would help me a lot if i could get some answers, thanks :)

It all depends on the context. Sometimes, you want to let the other strings ring, sometimes you don't.

Let's say you don't, notes that are fretted (not an open string) can be muted by releasing the pressure on the fretboard, just like you said. If the note is an open string, then you can mute the string with either your left (fretting hand) or right hand (picking hand).

Let's say you want to play the high e string (open 1st string) then play the 2nd fret 2nd string. If you don't want the high e string to ring when you pick the next note, you can angle your fretting finger a little so that it touches the 1st string, thus cancelling the ringing. You could also use your right hand to mute the string using your finger. So you would touch the 1st string with your picking hand fingers as you move to the 2nd string.

Hope this helps.